In principle, these days there are two classes of methods for reconstructing image data.
The first method class relates to analytic methods such as e.g. the so-called filtered back projection (FBP) method in the case of computed tomography projection data or a two-dimensional inverse Fourier transform in the case of raw magnetic-resonance data. Here, these respectively are single-step image calculation procedures, which are based on idealized mathematical system models that permit the derivation of analytic inverse solutions of the equation to be calculated. By way of example, if a record X of unknown object properties (e.g. local tissue properties that are ultimately represented by the image to be reconstructed) is assumed, these object properties are imaged on a record Y of measurement values by the system transfer function A (the so-called transfer function) of the imaging system:Y=A·X.  (1)
Here, the data records X and Y are written in the form of matrices. In such an idealized mathematical system model, both the transfer function A and the inverse A−1 thereof are known, and so, conversely, the record X of the unknown object properties, i.e. the image to be reconstructed, can be calculated as perX=A−1·Y  (2)from the obtained measurement values Y.
These methods are generally very time-saving reconstruction methods. However, the problem of these methods consists of the fact that in this case the non-ideal properties of real imaging systems are not taken into account. By way of example, in a real magnetic resonance system this non-ideal property can be seen in the always present Gaussian noise, in the inhomogeneous magnetic fields, in defects in respect of the recording sensitivity, radiofrequency inhomogeneities, etc. In the case of CT scanners, such real, non-ideal properties that should be taken into account during the reconstruction lead to e.g. quantum noise, scattered X-ray radiation, beam hardening effects, etc.
It is for this reason that relatively time-intensive, recursive methods are used in a second group of reconstruction methods, which recursive methods avoid the requirement of developing an analytic solution in advance.
Such an iterative reconstruction method usually begins with an initial estimate of initial image data X0. Image data established with a classical analytic method as per equation (2) may for example be used in the process. Using this initial image data X0 as a starting point, the image data is then successively improved iteratively as per the following equation:Xk+1=Xk−ΔXk  (3)
That is to say, image correction data ΔXk is calculated in each step k and used to generate updated, improved image data Xk+1 from the current image data Xk. The goal here is to minimize data mismatch between the real measurement values Y and the synthesized measurement values Yk, which are respectively calculated in the current iteration step as perYk=A·Xk  (4)
In order to calculate the image correction data ΔXk in each iteration step, use can be made of various algorithms, as known to a person skilled in the art from the prior art, for example the gradient of the steepest descent, a conjugate gradient or other similar methods. By way of example, the image correction data ΔXk may be calculated as per the equationΔXk=α·grad(fz).  (5)
The parameter α describes a relaxation parameter, which controls the convergence speed. In this case, it may be an arbitrary value, preferably in the range between 0 and 1, e.g. a value of 0.7. fz is a target function, such as e.g.fz=∥A·X−Y∥2.  (6)Here, the notation ∥Z∥2 is used for the L2 norm of Z.
The above-described method can also be expressed as a nonlinear mathematical optimization problem, restricted by boundary conditions, as per:min∥A·X−Y∥2,with X>0.  (7)That is to say, the target function fz as per equation (6) is minimized under the boundary condition that the values of the individual image pixels or voxels of the image data record X are all greater than 0.
Equations (6) and (7) hold true in the case of a so-called “least-squares minimization”. However, in principle, it is also possible to use other target functions, as will still be outlined below. The respectively ideal target function depends, inter alia, on the imaging system used.
The main advantage of these iterative methods includes also allowing the use of transfer functions A for nonlinear and non-ideal measurement systems.
In order to improve the iterative methods further and to take improved account of the real measurement conditions, it is possible, within the optimization method, to take into account a statistical model for the noise adversely affecting the measurement results. To this end, a method is described in, for example, Jeffrey A. Fessler, Statistical Methods for Image Reconstruction, John Hopkins University: “Short” Course, May 11, 2007.
To this end, a so-called noise regularization term is inserted in e.g. the target function fz in the above-described optimization method, which noise regularization term is formulated mathematically such that relatively high image noise is suppressed. Optimization methods using noise regularization terms are described, for example, in US 2010/0054394 A1 and in K. Zheng and J. Zhang, “Parallel MR Image Reconstruction by Adaptively Weighted H∞ Optimization”, Proceedings of the IEEE International Conference on Control and Automation, 2009, p. 1796-1800. The noise regularization term should preferably be constructed such that it is targeted to maintain the image contrast, more particularly edges and image sharpness, to the best possible extent within the optimization.
Additionally, for some time now there has been a novel reconstruction method, which is also known as a “compressed-sensing” method (CS method). This is a reconstruction method that can be used to generate quite good reconstructions from undersampled and/or compressed measurement data as well. The undersampled data is measurement data that was sampled below the Nyquist limit. It was found that under certain circumstances useful images can be generated from measurement data sampled under the Nyquist limit if there is random undersampling instead of coherent undersampling. By way of example, it is possible to record fewer projections, controlled in a random fashion, during each revolution of a computed tomography scanner. This leads to a reduction in the X-ray exposure of the patient. In the case of a magnetic-resonance system, it is possible, for example, for fewer lines to be sampled in an incoherent fashion in Fourier space, and so this can reduce the measurement time and, ultimately, the heating of the tissue by the radiofrequency load as well.
By way of example, a suitable reconstruction method is described in Michael Lustig et al., “Compressed Sensing for Rapid MR Imaging”, University of Stanford. The reconstruction of such undersampled measurement data is also carried out using an iterative method, with, however, the generation of an image with great “sparsity” being forced, which is why the method is also referred to as “sparsity enforcing reconstruction” (SER). In the process, an attempt is made to reconstruct the image from the undersampled data such that it does not have too many details. Depending on the posed question of the examination, this often makes sense. Thus, for example, in the case of angiographic recordings, the aim generally is only to visualize the contrast-agent-filled blood vessels to the best possible extent, with the details around the blood vessels being of little interest and rather bothersome. In images with too much image noise, even e.g. the noise forms details, but these offer no information for a subsequent diagnosis and are rather bothersome. Such cases also call for an image with relatively great sparsity.
By way of example, a sparsity enforcing reconstruction may be implemented by virtue of the fact that a so-called “sparsity regularization term” is utilized in the aforementioned optimization method, for example within the target function. Various options for forming sparsity regularization terms will still be mentioned below. By way of example, US 2009/0161932 A1 outlines a method in which two sparsity regularization terms are utilized side by side in the target function. US 2007/0110290 A1 describes a method in which use is made of a weighted sum of various sparsity regularization terms.